# Tagged Questions

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### perturbation theory solution of forced Duffing's equation

Question: Find the leading order of the asymptotic expansion for large t: $\frac{d^2x}{dt}+\varepsilon\beta\frac{dx}{dt}+x+\varepsilon x^3=Fcos(\frac{1}{3}\big(1+\varepsilon\omega)t\big)$ I have ...
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### Big-O Analysis: Max Bounded by the Sum

I have been asked to show that: $$\mathcal{O}(Max\{ f(n), g(n) \}) = \mathcal{O}(f(n) + g(n))$$ I have seen explanations of similar problems, but this is the first time I have encountered the ...
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### Calculate limit with factorial

I need to find the limit of this function..I thought about L'hôpital's rule, but can't seem to derive them both.. $$\lim_{n\rightarrow\infty} \frac{(2n)!}{(n!)^2}$$
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### Determine the realtions ($\mathcal{O}$,$\Theta$,$\Omega$ ) between $f(n) = \ln(n^{c} + n^{d})$ and $g(n)=\ln(n^{a} + n^{b})$

I am trying to determine the realtions ($\mathcal{O}$,$\Theta$,$\Omega$ ) between : $$f(n) = \ln(n^{c} + n^{d})$$ $$g(n)=\ln(n^{a} + n^{b})$$ Note: $a,b,c,d>0$ I need some advice how to use the ...
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### Need help in finding the asymptotic variance of an estimator.

I kinda doing some review questions for my finals and I kinda got stuck on this question. I'm able to do part a by finding the maximum likelihood estimator but for some reason. To find the ...
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### Leading order approximation to differential equation

Find a leading order approximation to the solution of $\epsilon y'' + 2 y' + e^y = 0$, $y(0)=y(1)=0$ as $\epsilon \to 0$. I know there is a boundary layer near $x=0$ and not at $x=1$ so I can ...
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### Prove or disprove the Big-O of an exponential function

$f(n) = 2^{n+1} = O(2^n)$ Intuitively, I think the statement is false. However, when I go about disproving it, I find that $2^{n+1} = 2^n \cdot 2$, meaning that if there is a constant $C$ larger than ...
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### Prove that $a^n$ is $O(n!)$.

I proved by induction that $2^n = O(n!)$. Can this fact be used to prove the following: Let $a$ be a positive constant and $n$ be a natural number. Show that $a^n=O(n!)$. I have already ...